Calculus Examples

$f (x) = \cos (3 x) \cdot \cos (3 x)$

Step 1

Raise $\cos (3 x)$ to the power of $1$ .

$\frac{d}{d x} [\cos^{1} (3 x) \cdot \cos (3 x)]$

Step 2

Raise $\cos (3 x)$ to the power of $1$ .

$\frac{d}{d x} [\cos^{1} (3 x) \cdot \cos^{1} (3 x)]$

Step 3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{d}{d x} [{\cos (3 x)}^{1 + 1}]$

Step 4

Add $1$ and $1$ .

$\frac{d}{d x} [\cos^{2} (3 x)]$

Step 5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \cos (3 x)$ .

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Step 5.1

To apply the Chain Rule, set $u_{1}$ as $\cos (3 x)$ .

$\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [\cos (3 x)]$

Step 5.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 2$ .

$2 u_{1} \frac{d}{d x} [\cos (3 x)]$

Step 5.3

Replace all occurrences of $u_{1}$ with $\cos (3 x)$ .

$2 \cos (3 x) \frac{d}{d x} [\cos (3 x)]$

Step 6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 3 x$ .

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Step 6.1

To apply the Chain Rule, set $u_{2}$ as $3 x$ .

$2 \cos (3 x) (\frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d x} [3 x])$

Step 6.2

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

$2 \cos (3 x) (- \sin (u_{2}) \frac{d}{d x} [3 x])$

Step 6.3

Replace all occurrences of $u_{2}$ with $3 x$ .

$2 \cos (3 x) (- \sin (3 x) \frac{d}{d x} [3 x])$

Step 7

Differentiate.

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Step 7.1

Multiply $- 1$ by $2$ .

$- 2 \cos (3 x) (\sin (3 x) \frac{d}{d x} [3 x])$

Step 7.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$- 2 \cos (3 x) \sin (3 x) (3 \frac{d}{d x} [x])$

Step 7.3

Multiply $3$ by $- 2$ .

$- 6 \cos (3 x) \sin (3 x) \frac{d}{d x} [x]$

Step 7.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$- 6 \cos (3 x) \sin (3 x) \cdot 1$

Step 7.5

Multiply $- 6$ by $1$ .

$- 6 \cos (3 x) \sin (3 x)$